Two Types Of Diffuse Predator-prey Model Analysis With Fear Effect

In nature,due to the direct or indirect interaction between the prey and the predator,the predator can kill the prey directly or produce fear effect on the prey.These two behaviors together affect the population size of the prey.In this paper,we investigate two kinds of diffusive predator-prey models with fear effect.The first one is a diffusive predator-prey model with fear effect and hunting cooperation.Based on the method of upper and lower solutions and the comparison theorem of parabolic equation,the global existence and dissipativity of solutions of the system are given.By using the principal eigenvalue theory of elliptic operators,we analyze the existence and local stability of constant steady-state solutions,and Hopf bifurcation.The second type is the diffusive predator-prey model with fear effect and mate encounters.The long time properties of the system are analyzed,including the global existence,dissipativity of solution of the system.The existence and stability of constant steady-state solutions,the Turing instability,and the existence and nonexistence of nonconstant steady-state solutions are given.Finally,the theoretical results are verified by numerical simulations.